College Algebra, 2013a

348 Rational Functions
the cost per system above and beyond the fixed initial cost of $150. Another way to interpret
our answer is that ‘infinitely’ many systems would need to be produced to effectively ‘zero
out’ the fixed cost.
Our next example is another classic ‘box with no top’ problem.
Example 4.3.5. A box with a square base and no top is to be constructed so that it has a volume
of 1000 cubic centimeters. Let x denote the width of the box, in centimeters as seen below.
height
depth
width, x
1. Express the height h in centimeters as a function of the width x and state the applied domain.
2. Solve h(x) ≥ x and interpret.
3. Find and interpret the behavior of h(x) asx → 0 + and as x →∞.
4. Express the surface area S of the box as a function of x and state the applied domain.
5. Use a calculator to approximate (to two decimal places) the dimensions of the box which
minimize the surface area.
Solution.
1. We are told that the volume of the box is 1000 cubic centimeters and that x represents the
width, in centimeters. From geometry, we know Volume = width × height × depth. Since the
base of the box is a square, the width and the depth are both x centimeters. Using h for the
height, we have 1000 = x 2 h,sothath = 1000 . Using function notation, 8 h(x) = 1000 As for
x 2 x 2
the applied domain, in order for there to be a box at all, x>0, and since every such choice
of x will return a positive number for the height h we have no other restrictions and conclude
our domain is (0, ∞).
2. To solve h(x) ≥ x, we proceed as before and collect all nonzero terms on one side of the
inequality in order to use a sign diagram.
8 That is, h(x) means‘h of x’, not ‘h times x’ here.